Representations and irreducible subrepresentations

[glmuelle]

I don't know how to do the following homework:

Let $$G$$be a finite group and let $$\rho : G \rightarrow GL(E)$$be a finite-dimensional
faithful complex representation, i.e. $$ker \rho = 1$$. For any irreducible complex representation $$\pi$$of $$G$$, show that there exists $$k \geq 1$$ such that $$\pi$$ is an irreducible subrepresentation of $$\rho_k = \rho \otimes \dots \otimes \rho$$ ($$k$$ times).

Hint: Let $$a_k = \langle \chi_{\rho_k}, \chi_\pi \rangle$$ be the multiplicity of $$\pi$$ in $$\rho_k$$; compute the power series $$f(X) = \sum_{k \geq 0} a_k X^k$$ and show that it is non-zero.


The definition of irreducible representation is that $$\rho: G \righarrow GL(E)$$ is irreducible if it has no subrepresentation which means that $$E$$ has no proper subspace $$\neq 0$$ that is stable under $$\rho$$.

$$\otimes$$ is the tensor product.

$$\chi_\pi$$ is the character of $$\pi$$ which is the trace of $$\pi$$.


I have no attempt at a solution because I don't even know where to start. For example, can someone tell me what the "multiplicity of $$\pi$$ in $$\rho_k$$ is? And what are the angle brackets? First I thought it's an inner product but since characters are scalars that doesn't make any sense. Many thanks for your help, I really appreciate it!

 

[Stephen Tashi]

I fixed the latex tags for you. Let's hope a group representation expert shows up soon.

(In case you didn't know this. The tag for LaTex on this forum is the "tex" inside square braces at the beginning and "/tex" inside square braces at the end. There is a bug with the way LaTex is cached. When you edit a message, you must use "preview post" and then, after the page appears, you must reload it with your browsers refresh button. If you edit a post you must do a similar process. Otherwise the LaTex looks screwy.)

I don't know how to do the following homework:

Let $$G$$ be a finite group and let $$\rho : G \rightarrow GL(E)$$ be a finite-dimensional
faithful complex representation, i.e. $$ker \rho = 1$$. For any irreducible complex representation $$\pi$$ of $$G$$, show that there exists $$k \geq 1$$ such that $$\pi$$ is an irreducible subrepresentation of $$\rho_k = \rho \otimes \dots \otimes \rho$$ ($$k$$ times).

Hint: Let $$a_k = \langle \chi_{\rho_k}, \chi_\pi \rangle$$ be the multiplicity of $$\pi$$ in $$\rho_k$$; compute the power series $$f(X) = \sum_{k \geq 0} a_k X^k$$ and show that it is non-zero.


The definition of irreducible representation is that $$\rho: G \rightarrow GL(E)$$ is irreducible if it has no subrepresentation which means that $$E$$ has no proper subspace $$\neq 0$$ that is stable under $$\rho$$.

$$\otimes$$ is the tensor product.

$$\chi_\pi$$ is the character of $$\pi$$ which is the trace of $$\pi$$.


I have no attempt at a solution because I don't even know where to start. For example, can someone tell me what the "multiplicity of $$\pi$$ in $$\rho_k$$ is? And what are the angle brackets? First I thought it's an inner product but since characters are scalars that doesn't make any sense. Many thanks for your help, I really appreciate it!

 

[glmuelle]

Thank you Stephen. I had tried to preview the post before I posted but somehow I got 'unknown error' three times. Then I just submitted it without previewing, I'm sorry for this.